How does a self employed person claim ESA?

Hi @madnurse and welcome. It is good to have you with us. I'm not an authority on benefits, but we do have them here and I am sure one of them will be along to have a word with you in due course, perhaps after the weekend.
For now you could probably do worse than have a look at the forum https://community.scope.org.uk/categories/benefits-and-finances. You could also possibly try the online benefits calculator here: https://www.scope.org.uk/support/disabled-people/benefits/check?_ga=2.118944524.405444417.1517054923-1832321402.1513208436
Hope that's of some use. Wiser heads will be along, so please bear with us and keep in touch.
Warmest best wishes,
Richard

Normally the simplest thing to do is to call and make the claim - the DWP will tell you if you're eligible. The number is 08000556688 unless you're in a Universal Credit area https://www.gov.uk/employment-support-allowance/how-to-claim