The minimum is set by the fund, not by the tax rules. The Opportunity Zone provisions in the tax code don't specify a dollar floor, so a fund could in principle take 10,000. What sets the floor in practice is the sponsor's economics. Every investor added means another subscription document, another capital account, another K-1 each year, and another set of distribution notices. A sponsor with 40 investors and a small raise spends a lot on administration per dollar raised, so minimums tend to sit at 100,000 or 250,000. Funds that accept 25,000 exist and usually run through a platform that handles the investor paperwork at volume.
The accreditation question is separate from the tax question. Most QOFs raise capital as private securities offerings, and those exemptions generally limit who can invest, which is why the fund page gates you before showing terms. That's securities law doing the gating rather than the OZ program, and whether a specific offering is open to you depends on the exemption it's using. A securities attorney or the fund's own counsel is the right place to confirm your eligibility for a particular deal.
The size of your gain matters for a different reason. You can only defer the gain amount, so a 40,000 gain and a 250,000 minimum means you'd be putting 210,000 of non-gain money into a vehicle designed around a tax benefit that money doesn't get. That's a real mismatch and it's the most common reason a smaller investor ends up in a fund that doesn't fit. Match the fund minimum to your actual eligible gain before you look at anything else in the deck.