IF there is somewhere writable, it is possible to save the current system time at shutdown...
Then during boot reload it.
The time still won't be correct - but it will be constantly moving forward.
This used to be done when/as the root filesystem got mounted, using the dismount timestamp for the current time.
I believe using "date +%m%d%H%Y.%S >shutdown.time" just before dismounting root (assuming it is writable), then using "date `cat shutdown.time`" should set the clock.
Of course, if you have a network connection using ntpd would work better. Assuming this was a RaspberryPI or something similar there is (or should be) a RTC available.
If you have a log file you could set the date based on that using the -r option (something like date -s `date -r /var/log/logfilename`, or maybe date -s --date="`date -r /var/log/logfilename`"), but you would have to experiment to decide if you need to get it in an acceptable format...
(BTW, the date -r uses the modification time of the file, and none of my examples have been tested for correctness.)