There are different types of atomic clocks, but in most common types, the output comes from a crystal oscillator, or more generally frequency synthesizer, which is then slaved to some spectral feature in the "physics package". That is to say, the phase noise is as good as that of the crystal in the unit, but the longer term frequency stability is much improved by the slaving.

An exception is an active hydrogen maser, which directly outputs the frequency of atomic transition. It has very good phase noise, but is a rare beast, which is only used where it is absolutely necessary.

It would be possible to build a hobby-class optical atomic clock using a quartz cell with iodine vapor and modulation transfer spectroscopy, which would be much more accurate than a rubidium clock, but it would be significantly more expensive. A iodine cell would be almost $800 and the rest of the components would add several thousand $ in costs.

Optical clocks are much more expensive than microwave clocks, because they need an optical frequency comb, which is a special kind of pulsed oscillator with a laser, to divide the optical frequency down to a frequency in the hundreds of MHz range, where you can use digital counters to measure time and frequency.

For a microwave clock, currently only 4 options are widespread, active or passive hydrogen masers, cesium clocks, rubidium clocks and clocks with mercury ions.

Clocks with trapped mercury ions, which can steer the frequency of an oscillator that provides a 40 GHz signal (typically after a frequency multiplication) are the most compact and reliable, but few hobbyists would succeed to build one. There are research articles that describe prototypes of such mercury clocks intended for use in satellites, which show how one could be made.

There is no option to make something cheaper than a commercial miniature rubidium frequency standard, unless you do some successful research and discover a completely new method.

Nonetheless, high-quality OCXO (oven-controlled quartz oscillators) are cheaper than rubidium clocks, and if used correctly they can be more accurate than miniature rubidium or cesium clocks.

For short time intervals, the rubidium clocks and the cesium clocks are no better than the quartz oscillators included in them, which are likely to be worse than a high-quality separate OCXO.

For times longer than a day the miniature rubidium and cesium clocks will have a lower drift, but you can achieve better than them if you compare periodically your OCXO clocks with good NTP servers and you create a model of your OCXO, measuring its aging rate and possibly also the influence of the ambient temperature.

After you accumulate enough statistics to characterize well your OCXO, even without Internet access you could maintain with it a more accurate clock and frequency standard than with a miniature rubidium or cesium clock.

This means that you would use a program to transform the accumulated ticks from the OCXO into time, taking into account the variation of its frequency with time and ambient temperature, modeled with low-order polynomials. Even using just linear dependencies would remove most of the OCXO error. Similarly, if you use it as a frequency standard, you would use a program to compute the current frequency, based on the current time and the current ambient temperature.