If my income is below $40,400 as a single person, is there a limit of how much stock I can sell to get 0% tax rate on my capital gains?

There is a capital gains worksheet that calculates the rate you pay on each portion of the gain.  But basically the gain gets added to your income and pushes you out of the 0% rate.  So only the first portion will be at 0%, then more will be at 15%, then the rest at 20%.

that's not how it works. use Taxcaster. you'll probably find you pay 15% to 20% on the capital gains in your scneario

This website has an easy-to-use capital gains tax calculator:

https://casaplorer.com/capital-gains-tax-calculator

For 2021, a single taxpayer with $40,400 in work income and an LT capital gain of $350,000 will pay a federal capital gains tax of $52,500.  

The calculator will also figure your state capital gains tax, if any.

"If my income is below $40,400 as a single person, is there a limit of how much stock I can sell to get 0% tax rate on my capital gains?"

Yes, there is a limit. How much capital gain you can have at the 0% rate depends on how far below $40,400 your other income is. The long-term capital gain gets "stacked" on top of your other income to determine the tax rate on the gain. But the $40,400 cutoff for the 0% rate is based on taxable income, not total income or AGI.

Here's an example, using 2021 figures. Suppose your total income, not including the capital gain, is $42,950. If you are single, under 65, and not blind, your standard deduction is $12,550. If you take the standard deduction and have no other adjustments or deductions, your taxable income is $42,950 - $12,550 = $30,400. Your long-term capital gain is stacked on top of that $30,400, so you can have up to $10,000 ($40,400 - $30,400) of long-term capital gain taxed at 0%. Any long-term capital gain above $10,000 will be taxed at 15%. A gain of $350,000 won't push you into the 20% bracket, which starts at taxable income of $445,850.

$40,000 income less the $12,550 Standard deduction = $27,450 taxable income. The22% tax bracket starts at $40,525 taxable income. This means the first $13,075 (40525 - 27450 = 13075) of any long term capital gain will be taxed at 0%.  The rest will be taxed  at 15% on up to $445,850.

In addition, if an individual has income from investments (including capital gains), the individual may be subject to net investment income tax.  individual taxpayers are liable for a 3.8 percent Net Investment Income Tax on the lesser of their net investment income, or the amount by which their modified adjusted gross income exceeds the statutory threshold amount based on their filing status. 

The statutory threshold amounts are:

• Married filing jointly — $250,000,
• Married filing separately — $125,000,
• Single or head of household — $200,000, or
• Qualifying widow(er) with a child — $250,000.

Reference: https://www.irs.gov/individuals/net-investment-income-tax