Can you prove, and with what assumptions, that a market will allocate some minimum amount of basic resources to every worker? I'm looking for a summary/less technical answer, as I'm not very well versed in the details of economic theory. I'm curious if there is some way to show that under some circumstances in a market economy that every worker (assuming they work some minimum hours, are not disabled, maybe have a basic high school education, etc) who participates in the economy will be able to procure some amount of basic resources (i.e. food, shelter, clothing).
More directly: In a market economy, will everyone who participates be able to sustain themselves, or can even a fully participatory, albeit poorly educated and unskilled, worker become unable to sustain themselves, and this is just an accepted fact of economic theory?
There is no guarantee that anyone in market economy earns some subsistence minimum. Generally, wage depends on marginal product of worker's labor and if there are some quasi-rents to be had also on the relative negotiating power of worker and employee. Of course, there is a bit more nuance to it, for that you can see Borjas: Labor Economics 5th ed, but assuming that people roughly get wage equal to their marginal product is reasonable first approximation.
There is no reason to a priori assume that marginal product of some worker will be enough to procure some basic goods and services (food, shelter, clothing are goods not resources - labor and other factors are actually resources in economics). So it might very well be that there are people for whom wages are not enough to be able to sustain themselves just on their own without some sort of subsidy/charity.