Work on projects of your own.
It’s tough, but you can learn enough to be a skilled data scientist even without any work experience, all from the comfort of your home. It’s going to require a tremendous amount of effort and inner motivation to do any significant work on your own. You won’t be financially rewarded, and the experience you gain by working on personal projects won’t count for much in the eyes of employers when you’re interviewing. In addition, you might need to work late evenings after you’re done with your day job or studies for the day. These are the types of sacrifices you need to make.
What I would suggest is to participate in Kaggle competitions. They expose you to many different machine learning techniques, and you get additional motivation since it’s a competition. It’s also a great place to learn from other data scientists. If you’re able to build a strong Kaggle profile and level up to let’s say Master, I can guarantee you that there are companies that will be willing to talk to you even if you don’t have any work experience.