It's all a matter of link budget.
Globalstar, for example, has satellites at 1400 kilometers, usable from a satellite phone with a low-gain omnidirectional antenna, as are Iridium's satellites at 780. Both provide narrowband to midband internet access. Inmarsat, Thuraya, and plenty of others have satellites in geostationary orbit at 35,786 kilometers that can provide narrowband-to-midband internet service to low-to-medium gain antennas (about 10 dBi, in the case of BGAN) by using large antennas on the satellite side.
Pictured: An Inmarsat I-4, above as deployed in GEO, below during assembly (humans for scale!)
These are, of course, L- and S-band services, which if your question is about Starlink, their satellites do not support (and are not licensed to use), but if it's a question of technical feasibility, it's very possible - there are plenty of companies already doing it.
If your real question is "What are the theoretical limits in user terminal antenna gain for a low-earth-orbit constellation providing broadband internet access", well, that's harder to say without a lot of details of the proposed network. However, since you talk about Starlink, if you assume the link budget of Starlink, we can do some math. This article puts Starlink's user-terminal-side received power at about -48.23 dBm - this is, by small-antenna satellite phone standards, obviously a very strong signal, but that makes sense - they'll need it for the data rates they want.
However, working from this number we can come up with a basic link budget. The antenna reflector on the aforementioned Inmarsat I-4 has a diameter of 10 meters. For mobile-satellite service, you generally want the lowest frequency you can get away with in order to keep required user terminal gain low, but realistically, such a hypothetical "MSS Starlink" would not be able to get as much L-band spectrum as it would need - lot fewer total MHz to allocate out down there. However, we can work with it. At L-band (1.5 GHz), the gain of the giant I-4 like satellite-side antenna should be about 43 dBi. From the same calculator, at Ka-band, 20GHz (the lowest-frequency MSS-allocated spectrum I reckon they could actually use for this), it's 65 dBi.
The distance from the user to the satellite is very much up to the constellation design. A satellite that must serve users at its horizon needs to support users much further away than its orbital altitude. However, since this is a "what is theoretically possible" question we can assume a very large constellation, in which the distance to the user approaches orbital altitude. I'll say, as somewhat of a guess, that max distance to the user is 400 kilometers, for a very dense constellation at your 340km orbit.
Plugging all that into here, and assuming a user terminal antenna gain of 4 dBi (this is pretty typical for an unpointed mobile satellite service terminal antenna) and adjusting the transmitter power until we reach our link budget, we find that the satellite would need to put out around 150-200 watts. This is high, but not utterly inconceivable for satellite communications. If we instead assume 10 dBi on the user terminal (BGAN-like gain - would probably need a small, few-element phased array to get away with this for a non-GEO satellite, but could still be a small flat panel, maybe on the order of 10x10 cm with like, a 2x2 array or so) we can get the needed satellite-side TX power down to 50 watts, more reasonable.
At Ka-band, thanks to the higher gain of the large reflector on the satellites, the link budget actually works out relatively similar, though one thing that calculator won't account for unless you calculate separately and add it in is atmospheric losses which are larger at Ka-band than L-band. Note that on the satellite side, due to amplifier inefficiency, the effective power requirement to transmit at Ka-band is likely 2-3x what it would be to transmit at the same power output at L-band. Getting 200 watts for one transmit beam out of a satellite in the Ka-band would be rather challenging, I would think.
I would also like to note that simply taking a GEO satellite and putting it in LEO is going to require some design changes - among other things, GEO satellites are not typically designed to have their view of the sun eclipsed by earth every 100-ish minutes, and while they do have batteries for when the sun DOES go behind the earth for them, this is much more infrequent and not as meaningful to their power budget. Adapting a GEO satellite to LEO, you would probably need more solar panels and more, possibly more and better (higher cycle-life - you'll see a lot more cycles on them in LEO), batteries.
Finally, the user terminal having such greatly decreased gain is likely to require network changes to ensure two satellites, even far-apart ones, do not serve anything close to the same area. With a high-gain steerable user terminal you get some degree of choice in which satellite you "hear" - not so at 4 dBi. This is a significant, if not insurmountable, consideration.
TL;DR: With enough of a satellite, it is possible. However, you're talking about taking some of the largest, most specialized MSS satellites in GEO and launching thousands of them to reach a suitably dense LEO constellation. It would be an exceptionally expensive undertaking, even on the scale of the already-high costs of GEO satellites or LEO constellations. According to the best info on it I could find, Inmarsat's satellites cost between \$100M and \$400M each. Economies of scale of course apply when producing more of them, but I think at Starlink-like numbers this could very easily be a trillion dollar project (while I'm sure someone will be quick to raise dropping launch costs, I would like to point out that for such satellites, launch, even launch of such large satellites to such a distant orbit as GEO, accounts for only a relatively small fraction of the cost to build, launch, and operate). On a purely technical basis however, yes, it is possible.
EDIT: Also, as for shrinking the user terminal down this small on the current constellation, no chance. No amount of redesign work will get the aperture efficiency of an antenna above 100%. They could maybe make it a little better, or make the modulation a little more efficient, but they're not going to go this far just with improved user terminals. You can't beat the Shannon limit and you can't get aperture efficiency above 100%.
