Can I still file my taxes from 2016 and still get a refund?

Yes as long as it is filed before April 15, 2020 or the 3 year limitations for refunds will expire for 2016 taxes.

2016 can only be printed and mailed.

https://turbotax.intuit.com/personal-taxes/past-years-products/

For federal taxes, https://www.irs.gov/businesses/small-businesses-self-employed/filing-past-due-tax-returns

doesn't say you can't. Be prepared to pay any penalties that might be owed, though the penalties aren't going to get any if you wait longer or get caught by the the IRS or other taxing authority (if you are also behind on your state taxes, the state(s) will want you to file too)

April 15, 2020 is the last day to file a 2016 return to get a refund.