Is there any way I can book separate airline's tickets on single ticket, without the help of travel agent? [closed]

If you are looking for a general way to "stitch" any two arbitrary flights onto a single ticket, then this is not at all possible (not even to travel agents), because the fare conditions for a given flight may prohibit it from being stitched with some others.

If you have a particular itinerary in mind, just do a search for this itinerary on a search engine, and it will give you all "reasonable" options, including itineraries which consist of flights of different airlines but can be booked on one ticket (because the fare conditions allow it).

Can I simply call Airline A1 and book ticket and then call B1 and book another ticket and ask them to "append it to the same ticket" as that of A1?

No; only the issuing airline can modify a ticket (modulo the occasional agreement between two airlines which have a very close partnership). You would need to call airline A1 and ask them to change your booking, if the fare conditions allow it. But you don't want to do that anyway; just book both flights at the same times to avoid having to rebook.

You seem to also be under the impression that a simple way to avoid booking fees is to "call the airline". This is usually not the case, and in fact many (most?) airlines will charge additional service fees for bookings made through their telephone reservation center (see British Airways).

A ticket from airline A can not be appended to a ticket by airline B, they will always be two separate tickets and two separate reservations.

Airline A maybe able to link the two reservations together, if airline B is in the same alliance, but it is still two tickets, so you would not get protected from late flights and such as a single ticket would. But it may let airline B know of your late arrival slightly earlier.

Airline A maybe able to book the flight on airline B for you, then issue a single ticket (or vice versa).

Travel agents and OTAs, like Expedia, do have the capability to stitch together bookings across multiple carriers, but some of the cheaper fares do not allow this type of booking.

I don't know of a publicly available website that lets you stitch fares and airlines together.

When you are searching for cheap fares, you sometimes have to give up something, like protection on tight connections, or build in longer layovers, eating up vacation time.

The answer is YES, but only for certain airline combinations.

What determines if this is even possible is the type of Interline Agreement the airlines have and if it allows them to book each other's inventory.

Most 'major' carriers have interlines, but not always, AA and DL for instance. Some airlines don't have any at all.

While a multi-carrier itinerary (which is different from a ticket) is linked (the PNR's, not the tickets) when booked, there is no way to link anything after the fact.