This is attorney Bill answering your question today. It is a pleasure to be of assistance to you in work-related visa question. The student visa does not allow off-campus work during your first year and thereafter permission to work off-campus is subject to permission by immigration to do so. After a year you can apply for CPT,OPT , permission because of economic hardship or working for a qualified international organization.
You make an application to USCIS to be granted permission to work off-campus.
This is the law on working on a F1 visa in the US. The SSN allows your earnings and taxes to be tracked during your stay in the US but does not confer automatically a right for you to work if it is off-campus.
That said a lot of people do run small businesses on ebay without a SSN, so while you may be able to actually run a business on the internet in the US, to be visa complaint you should first obtain authorization from USCIS.
Legally the business should also be registered in the US if the business is primarily doing business in the US with its main office there or if you are a sole trader, to be registered as a sole trader.
Obviously a lot of ebay sellers do business without being complaint with business laws nor visa requirements, but my answer is limited to explaining the law to to you.
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