In my antenna design course in graduate school, our Professor had insider information about the specifications the Starlink people were targeting for their mobile ground stations. He gave designing a similar antenna system as one of the final projects for the course. The exact numbers aren't important (nor are they really a tight secret; if you are well-versed in antenna design I imagine you could spend 10 minutes and come up with a list of specifications that are pretty close to what we were given in the course). We found that the mobile base station would be neither complex to construct, nor expensive (Conservatively <$100. We were focused on the antenna (with the right manufacturing deals we estimated ~$15-20 for total antenna fabrication costs) and we didn't investigate the radio or amplifier too thoroughly so we just budgeted $80 for those pieces and called it good).
The fundamental requirements were things like: at least X dB main lobe gain, electrically steerable to any point within a the cone 60 degrees off of azimuth, total physical size less than .5 m^2, operates at Y.YY GHz with resonant bandwidth of Z.ZZ GHz, etc... The design my team went with was an array of microstrip antennas with a network of phase shifters to electrically steer the signal via beamforming. It's actually quite amazing what you can do if you just get the right shape of metal to vibrate at the frequencies you pump into it. Antenna design is a wonderful confluence of mathematics, design and black magic.