Regarding this part of the question:
I've read that even the index funds with a low cost of entry still require a $1,000 minimum? Is there no way to get started in the stock market this year, even with only a few hundred dollars?
This is not true. There are many ETFs (Exchange-Traded-Funds) that you can buy shares in that cost < $1000/share. There may be mutual funds that require a minimum investment, but by no means are you limited to mutual funds when making an investment.
An example of an index fund you may want to look at would be VTI, which seeks to track the CRSP US Total Market Index. Essentially, it's an index fund that seeks to track the overall performance of the U.S. market, and it's trading at about ~$115.
Regarding this part of your question, however:
In fact, right now I probably couldn't even spare more than a few hundred dollars. But at the same time, with the stock market in a total free fall, I feel like some time this year will be an opportune time to invest that might not come around again for quite some time.
I would be careful with the "FOMO" of wanting to invest just because everything looks cheap right now, if like you say, you do not have money "to spare". There is no "Holy Grail" moment, and there will always be buying opportunities.
Investing should be looked at/carried out in a long-term manner, and if you need money quickly and do not have money to spare to invest, I would approach the situation with caution. As you can see, the market can be volatile, and if you have expectations of investing and pulling out your money quickly to fund some obligation or the other, you may want to take a step back and evaluate the situation.
Note: None of this should be taken as financial advice in any manner, and you should consult a licensed financial advisor before making any investment decisions.