Can one reach enlightenment if they need to care about the wellbeing of another? If one need to care about the well-being of another (physical health, mental health), one needs to constantly check their status to update information and act fastly. That makes one constantly worry. Therefore they cannot reach enlightenment. They can't even start meditation, even though one can temporary leave them a couple hours or even days to take care of yourself.

Is that correct?
A long time ago, before Gautama Buddha, in the time of Kassapa Buddha, there was a potter named Ghaṭīkāra. He was not a monk, but he was Kassapa Buddha's chief attendant. Puzzled about this, his student Jotipāla asked Ghaṭīkāra why he had not gone forth and become a monk given that Kassapa Buddha was so close by.


  MN81:11.2: ‘Dear Ghaṭīkāra, you have heard this teaching, so why don’t you go forth from the lay life to homelessness?’


Ghaṭīkāra answered quite simply. 


  MN81:11.3: ‘Don’t you know, dear Jotipāla, that I look after my blind old parents?’


Not everybody gets to be a Buddha's chief attendant. Ghaṭīkāra was special. And what Kassapa Buddha said about Ghaṭīkāra was also quite special. Kassapa Buddha said:


  MN81:18.15: He looks after his blind old parents.  And since he has ended the five lower fetters, Ghaṭīkāra will be reborn spontaneously and will become extinguished there, not liable to return from that world.


If you are a caregiver, you understand impermanence, love, compassion, rejoicing and equanimity. Take heart from Ghaṭīkāra's example and practice as you can, the Noble Eightfold Path lays open before you. Ghaṭīkāra was a lay non-returner. Perhaps you can be one too.

Who was Jotipāla? 

Jotipāla was Gautama Buddha in a previous life...that, too, is in the sutta.