Naively, this does not appeared to be permitted for holders of F-1 Visas.
M-1 Student Visa
The M-1 visa (Vocational Student) category includes students in
vocational or other nonacademic programs, other than language
F-1 students may not work off-campus during the first academic year,
but may accept on-campus employment subject to certain conditions and
restrictions. After the first academic year, F-1 students may engage
in three types of off-campus employment:
Curricular Practical Training (CPT)
Optional Practical Training (OPT) (pre-completion or post-completion)
Science, Technology, Engineering, and Mathematics (STEM)
Optional Practical Training Extension (OPT) M-1 students may engage in practical training only after they have completed their studies.
For both F-1 and M-1 students any off-campus employment must be
related to their area of study and must be authorized prior to
starting any work by the Designated School Official (the person
authorized to maintain the Student and Exchange Visitor Information
System (SEVIS)) and USCIS.
The issues are whether "employment" implicitly is limited to employment within the U.S., whether self-employment counts as employment, whether this is self-employment, and whether this self-employment (if it constitutes self-employment) is taking place in the U.S. for immigration purposes.
Resolution of some of these issues is very fact specific.
OPT at least, can be self-employment if it is related to your studies:
OPT Policy Guidance, section 7.2.1:
All OPT employment, including post-completion OPT, must be in a job
that is related to the student’s degree program. For students who are
not on a 17-month extension, this employment may include: (...)
Self-employed business owner. A student on OPT may start a business
and be self-employed. The student must be able to prove that he or she
has the proper business licenses and is actively engaged in a business
related to the student’s degree program.
So, if you are a history student, this is probably not allowed, but if you are a computer programming or business major, it might be allowed.
But, this comes with caveats:
In very limited circumstances, F1 students on OPT may be eligible to
qualify as self-employed business owners. Of course, the business must
relate directly to your studies. In addition, you must be able to
obtain any required business licenses.
You can read more about current US rules and regulations relating to
improving and expanding training opportunities for F1 students with
STEM degrees here.
In the most general of terms, F1 students are prohibited from
‘engaging in business.’ At the same time, students on F1 visa are not
expressly forbidden from ‘establishing’ their own business since they
are allowed to engage in ‘preliminary business planning.’
Again, it’s impossible to overstate that there’s a sharp legal
difference between ‘establishing’ a business and ‘engaging’ with or
working for it. The usual rule is that an F1 student is forbidden from
working for a company – even one s/he starts.
As described, there are some status modifications an F1 student can
make that would allow him or her to work for a business they
established. The CPT work authorization mentioned above is one way;
the other is OPT, also previously mentioned.
Would I be allowed to hire freelancers to work for the offshore
Would I be allowed to hire employees(US or non-US) in the offshore company?
There is probably nothing wrong with owning a company outside the U.S. that has non-U.S. employees or non-U.S. independent contractors to which you devote minimal attention.
If you have U.S. employees you are probably running a business without a visa authorizing you to do so within the U.S.
Does my presence in the U.S. as an F-1 student restrict me from
earning money to my home country company and home country account?
Nothing prevents you from sending legally earned money anywhere you want, or from earning money from property (such as passive investments) anywhere. It probably doesn't prohibit you from earning money from earned income for services provided in your home country either.
P.S: the business will have no presence in the U.S. other than the
owner living there and working for a job
If you are actually doing significant work while in the U.S. you are probably in violation of the F-1 visa subject to the narrow exception for OPT self-employment related to your studies.