The Verdict: Need More Info
Ireland has no general self-employment route; the only option is the Start-up Entrepreneur Programme for innovative, export-focused businesses. Check your business against its test, or come on an employment permit first.
It needs €50,000 and a business meeting the High Potential Start-Up test: innovative, aimed at international markets, and capable of ten jobs and €1 million in sales within three to four years. Success gives two years' residence, renewed on performance.
A conventional business serving Irish customers does not qualify at any funding level. If yours does not fit, coming on an employment permit and building the business later is slower, but it is a route that exists.
A parent or grandparent born on the island of Ireland would give you citizenship or a right to register, and the freedom to run any business.