It depends at what level you want to do 'recreational'. Do you enjoy wrestling the software, or reaching the noise floors? What do you want to tackle first, close to carrier, 1 kHz offset, far out noise floors, long term stability, Allan Variance, they will all need different tweaks to the acquisition hardware.
This is not the sort of project that numerous participants will write reams for on 'Instructables'. So you will be more or less on your own.
The PN3048 software used for your picture does appear to be available, in what's described as an incomplete (no Allan Variance), buggy, no-source version for windows (a few googles will find it), which might be able to process files without needing the original GPIB instruments to run. I would personally recommend writing your own in python/scipy/matplotlib, you will know what you've got, be able to hook up to modern scopes and SDRs, and be in total control of it.
If you rely on PC audio hardware with a simple mixer, then you're limited to the order of 10 kHz offset, and needing sources offset by a small amount, which is not what you get with OCXOs and GPSDOs. You might want to build your own 10.01 MHz source, to mix all your inputs down to audio, if you want to go this route. With standard PC hardware, you'll only be able to compare them pair-wise, though you can get external multi-channel audio inputs.
Although they have a poor noise performance, a multi-channel oscilloscope will be fine for capturing measurements for close to carrier and long term stability work, and at least getting your feet wet further from carrier. Digitise at 40 MHz, one 10 MHz source into each channel, downconvert and compare.
Software Defined Radios (SDRs) are a relatively inexpensive and (some) open source way of getting integrated RF and high quality fast ADCs into the hands of amateurs. You will find a large community of users online, though probably few pushing the low noise envelope.
Maybe signal could be compared to 2 independent OCXO and even better measurement extracted with very simple measurement setup.
Yes, it's possible to do two reference correlation measurements, though it only works spectrally, not for time based measurements. As you do multiple averages, the device under test noise grows as power, the self noise of the references grows as sqrt(power), so you get a 1.5 dB improvement in noise floor per doubling of measurement time, which means for any practical factor, it rapidly becomes time limited. I'd leave that facility until last, it's not a panacea for noisy references or hardware.