can you fill out a w4 to the extent you get no taxes taken out of a $ 50000 salary

Of course ... however if you are married, are the only one working and have 3 dependents under 17 then having nothing withheld would be correct.  

If you want more withheld ... on the W-4 choose the SINGLE filing status and/or  do not claim some or all of the kids on line 3  and/or  enter a flat amount on line 4C to be withheld on top of whatever is already being withheld.

Use the Tax Withholding Estimator on the IRS web site to figure out what you and your spouse  should each put on the W-4 forms that you give to your employers. It's much easier than using the worksheets on the form. It will let you adjust your withholding to get the approximate refund amount that you want.

Tax Withholding Estimator

. To qualify for this exempt status, the employee must have had no tax liability for the previous year and must expect to have no tax liability for the current year. 

Did you get a W2 or will you be getting one at the end of the year?  They could be paying you as an Independent Contractor on a 1099 and not taking out taxes.  Do they take out Social Security and Medicare tax?  

Basically, if you don't have sufficient taxes withheld from your W-2 pay, then not only will you owe taxes at tax filing time, but you'll also be assessed an underpayment/under withholding penalty on top of that. It would appear that's precisely what happened to you, based on your post at https://ttlc.intuit.com/community/investments-and-rental-properties/discussion/only-509-taken-out-of-a-w2-of-51000-and-received-underpayment-notice-from-irs/00/3075618