Many recent "looping" pedals theses days use 48kHz, 24-bit sampling. 16-bit would be a minimum to get decent sound quality. 10- or 12-bit? Maybe just for a toy experiment, or if you're after weird lo-fi sound effects.
A common, yet already oldish reference is the venerable Boss RC-1. It's spec'ed at 16-bit, 44.1kHz, and 12 min max record time. Uncompressed audio would require thus over 60MBytes. I don't know whether they use compression on this pedal or not; if not, a 64MBytes SDRAM chip would be adequate. Cheap stuff these days.
For the more recent 24-bit pedals, some use some kind of compression, either lossless or lossy to some extent, to extend record time. I suspect the 64MBytes RAM figure is a minimum on a recent pedal - probably on the cheapest ones. The higher-end ones may contain as much as 1GBytes or so. As I said, using a small DSP or MPU with an embedded SDRAM/DDR controller would make a pretty low-cost solution for this these days. It's become dirt cheap. Just look at the price of a RPi 4 with 1GBytes of DDR RAM.
Flash would be an option only if you want your looper to have non-volatile storage. It would require fast enough Flash (as far as write times and max access times are concerned), and would be much more expensive than a RAM-based solution. Now if some DIYers go for this solution, that's because it may look simpler to use a small MCU and an SDCARD for the job.
And yes, forget about SRAM. The largest, yet still relatively affordable SRAM chips these days are 4MBytes or so, and they cost like 4 times or more the cost of a 1GB DDR RAM chip. With 4MBytes, at 16-bit, 44.1kHz, uncompressed, you'd get approx. 12s record time. Forget about fancy compression algorithms unless you're ready to use a powerful DSP. MP3 *encoding* is processing intensive.
At this point, you could implement a looper with an MCU supporting SDRAM - maybe something in the Cortex M7 range, an audio codec and an SDRAM chip. Some MCU eval boards give you this, so that could be where to start IMHO.
You could look at using some kind of SBC instead, like the RPi. But the RPi doesn't have any decent audio codec on board AFAIK, so you'd need an external one. I guess an USB sound card should work, as it's likely to be supported by the common available Linux distributions. But then you'll need to figure out how to program audio on Linux (either using ALSA, JACK, etc...), and handle latency properly, as you'll get a pretty high latency with USB sound cards on Linux (I've managed to get low latency with these but only reliably with real-time patched Linux kernels, and there is none available for RPis that I know of.)
Note that a looper is not a sound effect, so latency itself is not an issue if you handle it properly - you just need to account for the latency when playing and recording audio, but that can be handled programmatically, as a looper only need to play back some already recorded audio, and record incoming audio at the same time, but doesn't need to process incoming audio in real time, so latency can always be compensated as long as it's known and fixed.