Disclaimer: I am not an attorney, and I have not 100% researched the law. Take any advise from an online forum with a grain of salt. Please consult an attorney, tax specialist, or the IRS directly for any concrete answers.
AFAIK there isn't anything that would prevent you from starting a business. Simply owing back taxes shouldn't make a difference on how you make money, whether that is working for yourself or someone else. All the IRS is concerned about at this point is that you still owe them.
When going through the process of forming an LLC a couple of years back, I don't recall any personal tax information being brought up except when we were discussing possible loan options.
Regarding loan options, one important issue you may come into is if the IRS has filed a lien against you:
A federal tax lien is the government’s legal claim against your property when you neglect or fail to pay a tax debt. The lien protects the government’s interest in all your property, including real estate, personal property and financial assets.
A lien will exist on your credit report for 7 years after it is released:
The IRS releases your lien within 30 days after you have paid your tax debt.
With a lien, it will be very difficult to get a loan or other financing for your small business. If this ends up being the case, you can try to get a discharge or subordination on specific property that would allow lenders a claim on your property ahead of the IRS. Otherwise, you may find yourself relying solely on what money you currently have.
A big point is the IRS's threshold on filing a lien is $10,000:
The Fresh Start Initiative increased the IRS Notice of Federal Tax Lien filing threshold from $5,000 to $10,000; however, Notices of Federal Tax Liens may still be filed on amounts less than $10,000 when circumstances warrant.
Since you currently owe ~$8,000 over the past 3 years, it is possible that adding another year in back taxes will cause the IRS to file a lien if they have not yet already done so. So it may be something to keep an eye on if you do plan on taking out a loan for your business.