Yes, but for a single person it tends to not be practical over the long-term. Many homebrew clubs actually do this to create their own yeast bank for members to have ready access to rare or specific yeast strains but it takes discipline, prep, and equipment.
To have any success in the long run you need to be careful about how you do it and follow certain procedures to maintain the purity of the culture so you might need some special equipment. But it will allow you to grow and maintain a house culture of yeast almost indefinitely.
Another option to stretch store-bought yeast much further but not to the level of a yeast bank is just to "wash" yeast between batches. That's simply a process of taking the slurry that you'll find on the bottom of your brewing vessel after racking off the primary fermentation, diluting it with distilled water in a beaker or jar, allowing it to settle for some time, and pouring off the dross on the top layer. What you're left with after washing yeast is a relatively active and pure culture that you can regrow a portion of into a starter or just repitch more must onto. The downside to yeast washing is eventually the culture becomes unusable because of yeast stress or other contamination, but you should be able to get 5-6 batches out of one store-bought yeast package which isn't anything to scoff at.