Most natural sources are broadband, so unless you are doing inerferometry, for total power radiometry, clock quality is not important, anything will do.
For hydrogen line and similar work, a TCXO will be a excellent, allowing Doppler analysis etc. Just for detecting it, a plain crystal oscillator, as found in most SDRs will be enough. Even then, if you want to start with hydrogen line, do not agonize much about the clock, there are other things, like antenna and LNA, which will be more important in the beginning.
It would be more important for narrow line sources like masers, especially if you work at 12 or 21 Ghz, you will need a better frequency reference, important for the down conversion, like at least a PLL LNB, if possible, hooked to a good reference (TCXO, OCXO).
Also, if you would want to study pulsar timing, you would need an even better clock, like GPS controlled.
Because amateurs aren’t likely to have a lot of aperture dedicated to their pulsar work, extracting the best SNR possible out of your system is really important. That includes better clocks, because otherwise, the pulses tend to get smeared down into the noise as you’re doing synchronous detection. Having at least a TCXO helps.