Sealing the inside of a water well to use it as a reservoir?
Your aphorism of the day: Theory and Practice are the same in Theory, but Different in Practice.

As a practical matter it's going to be near-impossible to fabricate, install, and install a pump or suction pipe into a plastic-sheet-tube without poking a hole in the plastic. If you manage to succeed, you've disabled your well - if no water can get out, no water can come in.

Generally, if you need a cistern, you'll do better to put in a cistern than to try and use a well as a cistern. 275 and 330 gallon food grade plastic pallet tanks are widely available for one quick/temporary solution, a stock watering tank with a fabricated cover is another that's widely available. If it's a long-term need, buried concrete tanks are the standard for large volume storage. If using the translucent plastic tanks, cover them to block light so you don't start growing algae.

Drilling your well deeper is probably the better solution. Alternatively, if you put in a cistern for delivered water, and you have some water in the bottom of the well, adding a small, slow pump to pump from the well into the cistern is another workable approach. While your main well pump presumably sucks the well dry when it's low, a much slower pump can extract a constant trickle of water and place it into your cistern/storage, from which you can pump to the house at higher speed when you need water.