A computer using an 8086 can provide memory protection by using an external memory management unit. This would be a chip or a circuit that sits between the CPU and the memory and provides an additional layer of memory translation, sends interrupts if out-of-range memory is accessed, and so on. I don't know if this was commonly done on the 8086 (I've never seen such a system described, but then I've not looked for one either), but was very common for workstations based on early revisions of the 68000.
(Edit: at least some systems were produced that used this approach, although as @RossRidge points out in the comments it was a little easier on the 68000 due to specific support designed into the processor, which is probably why it was more common there.)
For an 80286, the standard 286 protected mode provides all the isolation that you'd need to run a POSIX compliant operating system with memory safety.
(It wouldn't be a very good POSIX system, because memory allocations would need to be limited to 64K to fit inside segment limits, but POSIX allows for sizes to be limited as low as _POSIX_SSIZE_MAX, which is defined as 32KiB, so this is fine)
There have been a number of Unix-like operating system that run on the 8086 and 80286, including Minix, which is usually considered the forerunner of Linux (it is the system that Torvalds used at the time he developed the first versions of Linux and influenced the early development quite a bit) and Xenix. There is also a port of Linux to 16-bit systems called ELKS, although I don't know whether it supports memory protection or not (looking at the source suggests it probably does, but I've never really done anything with it so can't be sure).