One of the beauties of USENET was it’s store and forward model. As such I had no real issues using USENET and email back in the day with a 2400 baud Hayes SmartModem.
Also helped it was all lightweight text messages and, naturally, I didn’t download the entire feed, just the few topics I was interested in.
On the other hand, the 80386 and 68000 are both '80s era CPUs. The 80486 just barely squeezed in in 1989, with the fist systems available fourth quarter. Super VGA video cards were available by 1987. Macintosh system software had a TCP/IP stack in 1988. I would think that you could access modern social media from an '80s era computer if you went with the latter half of the decade. If you are talking about the first half? Not likely. A 1 MHz 6502 would have a hard time keeping up with SLIP over 9600 baud, let alone 56kb.