If you didn't have "min" it would be more like "well" is a physical object that you in this case couldn't feel, I don't feel my fingers, I don't feel the "well" etc.
When you're talking about feelings you have inside, as opposed to feeling something by physical touch you use "mi sentas min" like most Germanic languages do:
"Jeg kjenner meg"
"Jag känner mig"
"Jeg føler mig"
"Mi sentas min"
Eu sinto dor = Mi sentas doloron
Eu me sinto (sinto-me) mal = Mi sentas min malbone
I have to disagree with the other response a little. "Mi ne sentas bone" means that your ability to feel isn't good. The reason for using min is simply that senti is transitive. You have to say in the sentence what your feeling. In this case "I feel myself..." is analogous to the Enlgish phrase "I feel..." where it's implied that you're referring to yourself
Farti translates most closely to "to fare". It's the difference between "I am not feeling well" and "I am not doing well".