A ground station is going to involve some RF equipment first: an LNA to boost the received signal then that feeds into a down convertor to bring it to 1200MHz typically. The 1200MHz signal goes into a modem which can either be something very specific (think satellite TV set top box) or something very capable (think big server with a custom ADC capture card in the back). The modem will do some forward error correction. FEC can involve a mix of Reed-Soloman, Viterbi, and LDPC. The bits can now be sent out over a network connection. Uplink involves basically the reverse process except there is usually no FEC since you can always boost your ground antenna to be much more powerful than the one in space. An exception to that would be uplink signals meant to be relayed to the ground. More of those "bent pipe" satellites are becoming digital which allows them to process signals and make them smarter about channel utilization.

All of this equipment can usually fit within a rack or two. The downlink stuff can fit within a few U. Uplink involves some big power amplifiers. Back in the 60s it would have involved dozens of racks of equipment.