Generally No, because an FPGA has only limited amount of memory.
Yes, only if you specify the maximum amount of data you want to store in your memory.
Then you also need to define the memory topology.
FIFO coud be useful.
Again you need to decide: when FIFO is full, do you dismiss the oldest or the newest data?
As so often:
A development should start with clear requirements. With values and units.
A "bitstrem" could be anything from a morse code up to a gigabit telecommunications interface.
The whole idea is not clear. Draw a sketch about your idea. It tells more than a lot of words.