Write ADC data to memory and back to DAC (without MCU)
Yes, it is possible to do something like this. I saw a design from the 80's that used a ROM (27C64) and a 74HC4040 counter to generate a sine wave. The CD4017 counter also had one of it's upper address lines routed to an ADC that caused the ADC to sample and spit out a serial stream.
You could conceivably get an old ADC with a parallel output and use a counter to poll it and activate the WR lines of a parallel RAM, and then use a few flip flops to read the value back (I assume it's a delay you might be after)
But instead of building a physical system with 74HC series logic, the better way is to use an FPGA between the ADC and the DAC, especially if you don't need that much RAM, it would be much easier and more configurable.