I believe those are IBM 386 computers on the right based on the front of the machine, the megahertz display and the turbo button. The ones on the left are most likely 8086 machines. IBM brand I believe based on the front and the case design.
Several PC based BBS machines supported multiline operation without the need for mulitasking, though as modems got faster, supporting 16 modems on a single machine became impossible, and it was often cheaper to buy a new commodity desktop PC rather than a much more expensive machine with a 16-port serial card capable of handling the IO.
The machines might have been running a "DOS" flavour, but for the most part, the BBS software was the entire machine. Boot from DOS, but then what we consider DOS is effectively removed from RAM and the BBS software takes over.
The 386's might have been using DESQView or equivalent, and running a single line BBS as two separate instances on the same machine.
The machine would not boot either from network or harddrive. Network boot was mostly unreliable and a bad node on a token ring network would easily disable your entire ring. Internal harddrives would have been costly, power hungry and unreliable. I can almost certainly guarantee they booted entirely from floppy disks. There was no doubt a token ring network that kept them all communicating. A token ring could be wired to tolerate failure, but it drove the cost up.
Those are Miracom Courier modems on top on some machines, a mixture of 9600 and 14.4, and Hayes Smartmodems, on others, 2400 on the black & silver, 9600 on the all blacks. Almost certainly there are no 56k modems in that picture. And they would have almost certainly be plugged in to a dual port serial card that supports dual 16550 UARTs because the 16550 could handle up to 119kbps (118,500) IIRC (typing from memory, feel free to google it), and they had a 16-byte FIFO buffer so the CPU didn't need to poll or IRQ for data quite so often. Therefore they could support 56Kbps on two lines, though at that point, it would have probably be pegging the CPU with two users transferring files using a non-compressing XModem or ZModem.
Also, all the drives have been upgraded from five-and-a-quarter to three-and-a-half.