That particular style of spacedock is used a few times.
In each case the footage has been recycled from Star Trek III.
Starbase 74 (in orbit of Tarsas III)
Starbase 133 (in orbit of an unnamed blue planet)
Lya Station Alpha
(in orbit of an unnamed blue planet) Starbase 84 (in orbit of an unnamed yellow planet)
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Jupiter Station Jupiter Station , even if very likely smaller than the gigantic Spacedock-type starbases, seems large enough to have some internal docking space, at least for the smaller starship classes.
Starbase 173 Starbase 173 's design was based on the much smaller Orbital Office / Regula I stations, but it was evidently larger, here you can compare it to a Galaxy-class Starship: Starbase 375 Starbase 375 , with a design similar to the previous one, is shown here surrounded by various Starship classes.
Even this one seems large enough to have some internal docking area.
Unnamed Prison Base
The same design is used also for an unnamed prison base , that seems to be even larger than the previous ones:
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Utopia Planitia Fleet Yards is a dedicated fleet yard in orbit around Mars.
It's not explicitly a starbase, but otherwise appears to meet your criteria and does contain structures that would commonly be called a starbase in that it's a large installation in space.
There are some different perspectives in
The Next Generation and Voyager where the fleet yard appears on screen.
Here's Geordi LaForge during the TNG episode " Booby Trap " where he creates a holographic representation of Utopia Planitia and his girlfriend the Galaxy class designer Leah Brahms.
You can see through the window a shot of a Galaxy class ship under construction in what appears to be an expansive and enclosed space compared to the typical spacedock we've seen in other episodes and films.
In the Voyager episode " Relativity ", we actually get a wide shot of the Utopia Planitia Fleet Yards as a whole.
Here, we see our typical spacedocks in the foreground, but there are some much larger stations in the background, including the station that is presumably the basis of the holodeck program that LaForge ran in "Booby Trap".
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Starbase 47 , from the Vanguard novel series, is a large starbase that lies in the Taurus Reach, a narrow band of space between the Tholian Assembly and the Klingon Empire.
It is also of the 'enclosed' kind, capable of holding up to four Constitution class starships inside.
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In extended canon (as in spec books and such), there are at least all these shipyards in addition to those mentioned above.
They would all have repair capacity as well as construction and living quarters at a minimum.
Some repairs take a very long time or require external life support so it comes as standard.
Assuming that we are talking about prior to the arrival of the Borg which would destroy a number of these.
40
Eridani A Starfleet Construction Yards, Acamar Prime Shipyard, Aldea Prime Shipyard, Altair Prime Shipyard, Andoria Prime Shipyard, Antares Ship Yards, Barcon Prime Shipyard , Barisa Prime Shipyard, Beta Antares Ship Yards, Bolarus Prime Shipyard, Calder Prime Shipyard, Centauri Spadeworks, Cestus Prime Shipyard, Copernicus Ship Yard, Deneva Shipyards, Denocet Prime Shipyard, Dytallix Prime Shipyard, Earth Station McKinley, Erabus Prime Shipyard, Finnea Prime Shipyard, Galor Prime Shipyard, Gavor Shipyards, Gibraltar Shipyards, Idran Prime Shipyard, Invernia Prime Shipyard, Iyaar Prime Shipyard, Janus Prime Shipyard, Ledonia Prime Shipyard, Meldrar Prime Shipyard, Merak Fleet Yards, Minos Prime Shipyard, Morena Fleet Yards, Moriya Prime Shipyard, Nausicaa Prime Shipyard, Oyarsa Prime Shipyard, Proxima Maintenance Yards, Qualor Prime Shipyard, Riverside Shipyard, Salazar Shipyard, San Francisco Fleet Yards, Tau Ceti Prime Shipyard, Taurus Prime Shipyard, Tellar Prime Shipyard, Tessen Prime Shipyard, Tranquility Base, Tri-Rho Nautical shipyards, Utopia Planitia Fleet Yards.
Now the manufacturing is out of the way, lets look at internal ship accommodation.
If we move to starbases with internal docking like Earth's one, you'll find that it's a 'cookie cutter' external design that is scaled up or down to fit needs (according to the Tech Manual) called the Spacedock-class.
Designated classes will tell you what does what.
Starbase 2 for example is a K-class with a general design that looks like this:
Note 3 arms coming from the centre.
This is Starbase 343, but like all it bears K7 markings as DSS K7 was the first of it's design.
K-class still can dock cargo ships internally, but hasn't got the space for a ship like the Enterprise or even Akira class.
It will happily take quite a few Sabres inside though, or 2 trade vessels.
Part of the reason that trade ships were long and thin were to fit into K-Class and other smaller starbases.
Back to the Spacedock-class, there are only 5 built.
Every other one is a military installation that flat-out OP's anything but a direct armada assault.
40 phaser banks huge result capacity and ability to hide a large fleet etc etc.
Earth one doesn't have guns because politics but you already know that.
Finally, it was said earlier that the space dock at earth was grown in size for video after new bigger ships arrived...
well officially, the doors are resized on all Spacedock-class bases whenever a larger warship is made.
For a long time these were the only non-shipyard facilities that could internally dock warships.
Anyway, the full list is as follows: Lya Station AlphaSpacedock, EarthStarbase 74Starbase
84Starbase 133As for other classes a few come to mind:
The Sigma/R1 class starbase would have docking facilities for cargo ships internally.
You don't actually see any of these until the 24th century and are refitted to Sigma class, but the R1 class of the 23rd was visually the same allegedly.
Note that the Nor class has 3 'stalks' and the Frontier has 4.DS9 is a Nor-Class Starbase which had many clones too, as it was originally called the Terok Nor under Cardassian possession.
There is also the Elvok Nor and Empok Nor which are almost visually identical.
It's replacement was a Fed Frontier-Class staircase which again is one of the only other large military installations the Fed have possessed.
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There appears to be at least 2 in orbit around Earth.
However, this image is from the non-canon game: Star Trek Timelines .
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In the Star Trek universe, there is a massive star base (one that can house several ships inside of it) orbiting around Earth.  Is this the only one of it's kind in the Federation, or do other planets have similar stations near it?  I understand that there are ones like K-7 and DS9 that have ships that dock, but I mean large enclosed ones that is similar to the one around earth.
Did the Federation have any large starbases orbiting other planets?
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It is possible that the Federation had other large starbases orbiting other planets. The Jupiter Station, for example, should have had enough space for some internal docking space. Similarly, Starbase 47, from the Vanguard novel series, is a large starbase in the Taurus Reach. There is no definitive answer on this regarding starbases specifically, but it looks like Jupiter Station and Starbases 173, 375 and 47 were big anough.
No clear answer to this question, but it looks like Jupiter Station, Starbase 173 and Starbase 375 were big enough to house Yes - Starbase 47 is big enough to hold up to four Constitution class starships inside.