No you don't have to be super-rich. But... the companies do not have to sell you shares, and as others mention the government actively restricts and regulates the advertising and sales of shares, so how do you invest?
The easiest way to obtain a stake is to work at a pre-IPO company, preferably at a high level (e.g. Director/VP of under water basket weaving, or whatever). You might be offered shares or options as part of a compensation package.
There are exemptions to the accredited investor rule for employees and a general exemption for a small number of unsolicited investors. Also, the accredited investor rule is enforced against companies, not investors, and the trend is for investors to self-certify. The "crime" being defined is not investing in things the government thinks are too risky for you. Instead, the "crime" being defined is offering shares to the public in a small business that is probably going to fail and might even be a scam from the beginning.
To invest your money in pre-IPO shares is on average a losing adventure, and it is easy to become irrationally optimistic. The problem with these shares is that you can't sell them, and may not be able to sell them immediately when the company does have an IPO on NASDAQ or another market. Even the executive options can have lock up clauses and it may be that only the founders and a few early investors make money.