I think making a character seem real enough to empathize with requires a two-pronged approach.
One, the character needs to care about what's happening to them. They might be struggling with something fundamentally petty and unimportant, but it the character is convincingly distraught about it, the reader will empathize with their pain.
Two, you need the character's goals and/or actions to be something the reader can relate to, understand, and even approve of, given the character's circumstances.
For example, the character Dexter is a psychopathic serial killer, but the reader knows he (1) only targets evil people (and who can't relate to that?) and (2) Dexter is genuinely trying to be a good husband and dad, despite the fact he has no emotional capacity to be.
So Dexter, who is really quite an appalling creature, is empathizable because we can sort-of approve of his actions, and his struggles are both poignant and worthy.
In summary, make sure the audience thinks the character is a worthy person, and then make sure you show that the character cares very much about what's happening to them.
(i.e. If they're dying, show their pain and terror. If they got a paper cut and stained their shirt, show their crippling anxiety at having to give a presentation to the boss in a blood-stained shirt.)