Electronic – How to clean a power rail for sensitive ADCs

You'll need a switcher to pump up the input voltage, and have it followed by a linear regulator, like Russell proposes. His TPS717xx, however, while having a high PSRR doesn't have a high output current (150mA). High current and high PSRR don't go together well on LDOs. Nevertheless, an LDO may be required, since the 2-4V input volatge suggests that you're powering from batteries, and then you don't want to lose too much power in regulators.
A solution for the PSRR may be to use 2 LDOs in cascade, so that the PSRRs add.
The LP3878-ADJ may be a good choice for your second stage; it's low-noise, can supply 800mA and has an adjustable output which can be set to 5V.
For the preceding stage you could use an LP38690-ADJ. The datasheet shows how you can improve ripple rejection at higher frequencies if you place a 100\$\mu\$F output cap. Caps between each stage (100\$\mu\$F + 10\$\mu\$F + 1\$\mu\$F) are a must, but \$\pi\$-filters are a better idea: caps + ferrite bead + caps. Those should suppress the higher frequencies generated by the switcher.

edit (re your update)
Dave rightly has second thoughts on your using AA batteries, but it's even worse than he calculated. I estimate that you'll need 7V from the switcher to feed the cascaded LDOs, so 600mA @ 7V is 4.2W, at a conversion efficiency of 80% that's 5.25W from the batteries. Two AA cells deliver 3V, so that's 1.75A from the batteries! Not only won't they last long, they won't like it either. I suggest you use a LiPo cell, which delivers 3.7V with a boost regulator, or even two cells in series, giving 7.4V, and use just the LP3878-ADJ. (If you don't use a switcher you don't need the second LDO to suppress its ripple.)

Your piece of art is most likely worse, noise-wise, than a breadboard would be.

Noise is specified over a certain bandwidth, and so is the LM317's PSRR:

The graph shows indeed a >60dB rejection for 100Hz-120Hz, which is the typical power supply ripple frequency from a bridge rectifier. That's what the LM317's ripple rejection focuses on. For higher frequencies (>100kHz) the ripple rejection is pretty bad, at levels of a mere 10dB. If much of your noise is in that band it will barely be suppressed by the LM317.
So what can you do about it? Connect capacitors to the LM317 input and output, as close as possible to the pins. I would place 1\$\mu\$F parallel to 100nF. Do the same for the opamp's power supply pins: two caps as close as possible to the pins. The 100nF should be the most close. This should substantially improve your noise figures, and you'll see this on your scope. BTW, reading your noise from a scope as a peak-to-peak value is not the proper way to measure noise \$-\$ you should measure the power over a given bandwidth \$-\$ but it gives you a first idea.

The \$C_{ADJ}\$ mentioned in the graph, which shows a >10dB improvement in ripple rejection should be placed parallel to your 710\$\Omega\$ resistor, as shown on page 16 of the datasheet.