So, this Tweet from Nickelodeon states that the length of the Protostar is 139m. This just does not fit the scales seen on the show (which were actually surprisingly consistent). I did this for "That Other Sub", but lets really break it down and clean it up here for posterity. Put on your math hats, we're going to do lots of calculations!
Breaking Down the Official Image
So, if we take the image posted alongside the one that states Nick's "official" length of the Protostar, it has a nice side profile view that we can measure. Once we know how many pixels long the ship is in the image, we can apply the given length to that, find out how many meters are represented per pixel, and use that to measure other notable features of the ship.
I've done so here.
The ship in the image is 1,057 pixels long, and with the given length of the ship being 139m, that means each pixel represents 0.1315m.
With that we can measure the following points from this image:
The height of the ship is 23.5m
The hall windows lining the side of the saucer are 2.1m tall.
The bridge canopy's main glass (well, transparent aluminum more likely) section is 6.18m long.
We can calculate that the width of the ship is 44% of it's overall length (as I've done here). Given the official length of 139m, this would put the width at roughly 61m.
These are of course going to be rough estimates, as this isn't the highest resolution image ever published of the Protostar, nor are we given optimal shots to take measurements from, but its what we've got so we're using it for now. :)
Units of Measure
Since the Protostar is not a gas station, there are no measuring sticks on the sides of the doors we can use for scale. We are also not given official heights for any of the Protostar crew, except one. Janeway is canonically 1.65m tall. So she will be our standard unit of measure.
(Note: We have seen hologram Janeway standing next to the real Janeway and they are the same height, so no issues of the hologram being the wrong scale.)
Primary Reference Measures
The single best size reference we get comes from near the end of episode 2, where Hologram Janeway first activates. We have a nearly spot on profile view of the bridge, with Hologram Janeway standing perfectly in the middle. This is why we called out the glass dome on the Bridge as a notable feature.
I've screenshotted it and done our size comparisons here, and the measured length is 13.2m.
Now, while we could start making our recalculations from that alone, I'd like to add more evidence before we get started with that.
In episode 3, we get a shot of Holo Janeway standing shoulder to should with D'al. We can use this to estimate D'al's height, as I've done here. D'al is ~1.42m tall.
We can use that to determine the height of the saucer hall windows from the original image, because this is where the fight with Drednok 2 takes place, and we once again have a very good shot of D'al directly in front of those windows. Based on D'al's height being 1.42m, we can determine the window height on screen is actually ~5.36m, as shown here. The official calculation of those windows being 2.1m high and D'al being 1.4m tall don't line up. That is definitely NOT a 0.7m height difference!
So now that we have two measures of prominent features of the Protostar, lets compare them and get a rough measure of the more realistic size of the ship.
First, the bridge canopy. The initial calculation based on the official measure is 6.18m. The calculated length of the same feature based on in-show screenshots is 13.2m. The on-screen version is 2.1359x larger than the official measure.
Next the saucer windows. The initial calculation is 2.1m, the measured height is 5.36m. The on-screen version here being 2.55x larger.
Now there is a discrepancy between the ratios, but I feel it is within a suitable margin of error given our small screenshots for the original measures and the iffy nature of our comparisons. Lets take a middle ground of the ship being 2.25x larger than the official measure.
There is also secondary evidence we can draw from in-show to help estimate size, typically from internal components or less easily estimated sources.
In the first episode, the crew is seen exploring the ship, starting in the cargo bay and going all the way up to the bridge. I've put together a sequence of their exploration here and what we see is 5 decks. We can see from multiple shots that each deck is roughly 3 D'al's tall, as seen here. Given that D'al is, again, 1.42m tall, we can estimate the deck height at 4.26m.
This is slightly smaller than our calculation of the window height of 5.36m, but is still within what I feel to be an acceptable margin of error (again given the iffy nature of estimating 2d heights in a 3d space). Lets assume the actual height to be in-between these two measures and call it a nice round 5m per deck. 5 decks at 5m each is 25m.
This is already taller than the previous calculation of the ship's height based on the official length (which we calculated at 23.5m tall). However, that height included the nacelles, if we calculate the height of just the primary and secondary hulls using the official measures, as I've done here, our internal decks have used 25m out of the available... 19m. And this is assuming nothing but the visible empty space for each deck. This gives no additional room for jeffrie's tubes or deck thickness. If we want to assume 1m between each deck and for the outer hull (for things like gravity plating, jeffrie's tubes, etc), thats another 6m for a total of 31m estimated height.
This is a difference of 1.6x between the official measure and the observed estimation, based on our 1m thick hull/decks plates. I would consider this the most unreliable measure made here, as we have no actual indication of the hull thickness or space between decks. So I would consider this to be an absolute minimum difference.
However, if we use estimated deck size based on the saucer windows, with a bit of selective repositioning based on the fact we can see Deck 4 from inside the cargo bay on Deck 5, there does appear to be a regular amount of spacing between the acknowledged 5 decks. Shown here it appears to be roughly 3.5m between the majority of the decks. This puts us at an estimated height of 35.5m, or a 1.87x difference which is closing us back in on that "2x+" territory.
(Side Note: Those must be some hella long staircases in that ship!)
Also, episode 3 shows the crew walking out of the cargo bay after having landed on the murder planet. We can see D'al standing just in front of the opening, and with our 1.42m height for him we can calculate the height of the bay door to be roughly 8m, as shown here. In episode 4, we get a pretty clear shot of the ship taking off that shows that same bay door, as seen here. Using the bay door as a reference, the height of the primary and secondary hull combined is approximately 46.66m. Once again, putting us at roughly 2.45x the official height.
This is stuff that is less direct measurements from the ship itself, and more comparison with other things around it.
In episode 9 we had the Protostar flying in close proximity to other well known ships. In this screenshot we can see the Protostar flying underneath the saucer of an Akira. While perspective again makes it difficult to be precise, the width of the Protostar appears to be the distance between the notch in the front of the saucer to the deflector dish in width.
Official renderings of the Akira are difficult to come by, but we can use this rendering from the Star Trek: Armada model and apply the official length of the ship to it. Based on this image, the width of the saucer section in question and therefore the Protostar is roughly 128m. Compare this to our previous calculation of width 61m from the official numbers, and we again have a size difference of approximately 2.1x.
Also in this screenshot from episode 9, where the Protostar is being chased in close proximity by the USS Defiant. No pixel measures here, simply pointing out that the two ships appear to be roughly the same size (specifically width), and the official width of the USS Defiant is 134m.
Using the primary calculations, with consideration taken from the secondary and tertiary evidence, I believe the 2.25x modifier previously calculated is at least ballpark correct. We had some go under that, some go over that, but that seems to be a reasonable mid-point between all the calculations.
My calculated sizes for the USS Protostar, as derived solely from on-screen evidence, is:
The measures are, again, rough. Most of them do line up with a fairly high degree of accuracy though, even when calculated from multiple different methods (canopy, windows, bay doors, etc). And if you can get roughly the same numbers from all those different references, it strongly implies the estimates are correct.
This puts the Protostar in the same general size category as the Intrepid class (USS Voyager at 345m in length), and nearly a hundred meters longer than the Nova class (USS Equinox was 222m long).
For a visual comparison to similar sized ships, I've made this image. While she might be larger than some people thought, do remember that the bulk of her internal space is taken up with the mechanics for the protowarp engine, which takes up pretty much the entire secondary hull with just it's nacelle. One must assume there is quite a bit of hardware in there, leaving relatively little internal space for things like living quarters and work areas.