Hypothetically, if I owned a private airstrip on my property, with all the necessary permissions to do so, and I only want certain aircraft landing there, would I be allowed to issue takeoff and landing clearances even though I am not an air traffic controller?
An uncontrolled field is just that - uncontrolled. No pilot will accept a clearance in uncontrolled airspace in the first place. Consider the legal consequences if two aircraft were to collide as a result of a "clearance" issued by you. Since the aircraft are in uncontrolled airspace, it is the responsibility of the pilots to make sure they do not get too close. However, if you got on the radio and started issuing "clearances", you might trick pilots into thinking that you are in control. If something bad happens, you should not expect to just walk away from it.
Besides, what makes you think you are qualified to provide air traffic control? Becoming an air traffic controller takes years of intense training, and only the top few percent of applicants are admitted to the education in the first place. Even the fact that you have to ask this question makes it seem like you have a very poor understanding of air traffic control.
The issue of not wanting people to land on a private airstrip has nothing to do with air traffic control. Air traffic controllers do not exist to tell pilots what they can and can't do - they are there to provide a service to the pilots! As others have pointed out, this would be a matter of trespassing. Pilots are very well aware that private airfields are just that - private. No one will just randomly spot a private airfield from the air and decide to land there without having first done some research - unless in emergency situations, of course.
In the past when airport owners want to control traffic flow, they file a NOTAM that the airport is closed, and contact xxx at 123456789 for exceptions. Then for the takeoff, the airport manager would issue a verbal "the airport is open for N1234 to depart" and then close the airport. This could be done on the unicom, in person, via telephone or however one wanted to. This happens at some private and corporate strips. It is actually more common in the northern parts of Canada, where there are fields operated by mining companies. In those cases, authorized aircraft come and go, and the field appears as a restricted use field. (Often they are not even charted, even though they may have VOR-DME on the field.)
So in summary, you have at least two paths to take: 1. NOTAM the field closed, or 2. Restrict the use
If you really want to play controller, you could install a private tower, or Non-Federal Contract Tower (NFCT), but then there are a bunch of rules to follow, and I believe your tower controllers would essentially have to be FAA trained. But I am not up on those regs, and they may have changed since I was more fresh on them. Here is a sample NFCT set up for an event.
Specifically, the FAA permits NFCTs, whereby an agency or entity that wishes a towered airport which does not meet federal criteria, or is otherwise unable to get an FAA Contract Tower (FCT), may operate a control tower.
Technically, this satisfies the OP criteria for issuing "clearances" but perhaps with a little more red tape than he might be interested in.
AC 90-93B states in part: "There are no comprehensive set of federal rules governing the operation of NFCTs." The Advisory Circular goes on to make recommendations for procedures, policy, training and equipment.
In summary, if you want to install a tower at your private field, you can do so. Good luck getting insurance. Make sure you follow FCC Part 87 for your radio licenses. Spend some time working with your center, and regional approach facilities.
Without a control tower, the field would be "uncontrolled" in the eyes of the FAA, so no, you couldn't issue clearances per se. The FAA would probably take a dim view of an untrained individual playing air traffic controller at his private field.
That said, the field would still be private property, and you'd be within your rights to bar anyone whom you didn't invite from landing there. "No tresspassing", etc. I don't think very many pilots assume that they can just land at any private airstrip they see -- emergency situations notwithstanding.
As a practical matter, even if you built yourself a platform & got a radio, those whom you hadn't invited would probably have no way to know what frequency to select, in order to be told not to land there. I don't think private strips get UNICOM or CTAF frequencies published on the charts, mainly since the traffic at such fields is presumed to be very, very low.
There is also AFIS (aerodrome flight information service, callsign Info) and Radio type stations. These are not control towers and operators are not air traffic controlers. They provide advisory services and information, they do not control air trafic.
It is much more easy to get one of these stations to a small airport and its ATZ - aerodrome traffic zone. The people providing this service are indeed often owners or members of the club owning the airport.
They can tell the pilots if the runway is free for takeoff or landing, but they do not issue clearances you for that. They tell the pilots about other traffic in the circuit, in the ATZ and its intentions. To some extent you can this way get to similar activity as tower ATCOs, but your instructions are nor clearances, they are advisories.
AFIS is a little bit more involved. For example in Czech Republic most small airports had AFIS but almost all of them were converted to Radio which has fewer requirements because the common European regulations for AFIS and AFISO (AFIS officers) became more strict.
An airport can be controled in certain hours and uncontroled with just an AFIS at other times (LKLK).
The FAA has thousands of locations listed that are uncontrolled private fields. Some offer services, some don't, and some don't even have safe landing strips any longer.
They all have an ATM, and probably at the time of the submission had everything required for being a non FAA controlled Airport. The requirements have likely increased over time. (I've contacted the FAA about strips where the ATM has died and was told they don't have the manpower to check the small airstrips.)
You would in conjunction with the local ATS be able to approve and issue landing clearances for your airstrip. That is why an airstrip plate has the Airport Manager and Phone number listed.