I will be using 1:4 gear ratio and thus will require a motor with a continuous 360 motion and a high torque for efficient functioning. Servo motors with continuous rotation and high torque are very expensive and rare, so my choice has boiled down to DC Motor and stepper motor. However I have not completely understood the way to configure them for a robotic arm. I don't want to use rotary encoders, as installing a potentiometer or hall effect sensor on the shaft will be cumbersome and inaccurate due to vibration. Is there a way one could move a DC motor or stepper motor to a particular angle with IC/modules like l293d or other motor drivers?
You can't control shaft angle of the DC motor without any feedback from it. Servos made for RC models or robots just do the thing - they utilize a potentiometer or rotary encoder to measure shaft angle and control it, mostly with PID regulator.
You can utilize stepper motor for Your task, as these motors are mainly used in the angular manner of control. For that, better than L293D will be dedicated stepper motor driver, like A4988 or DRV8825 (I don't know how large Your robotic arm will be, so these are just example drivers).
If You will choose DC motor over the stepper motor, mind that servos' mechanisms have much higher gear ratio than 1:4 in order to gain torque from a small DC motor.
EDIT: There is also a problem with holding the position of the stepper motor, common in CNC machines, 3D printers etc. Stepper motors can move (almost) freely, when there is no holding current applied. Therefore, when a machine is turned on, every motor has to find its origin. It is usually done by placing a switch on the minimum or the maximum position of the joint. The process of finding origin of the machine is called Homing.
Check out here how it is done by a 3D printer- You can see here a movement of the axis towards the switch. When the switch is hit, it is recognized by the software as reaching the minimum of the axis, thus zeroing joint's position.
Moreover, there is also a way of sensorless homing by measurement stepper motor current draw. For example, TMC2208 (and also TMC2130 and TMC2100) stepper motor drivers are capable of doing that. When the joint mechanically reaches its constraint, the driver knows it by measuring, that the motor is drawing more current than normally. It also sets
diag1 pin to high (if it is properly configured), allowing external electronics to detect machine's home position.
These drivers are also very silent and give smoother motor movement in comparison with previously mentioned ones. You can check the difference in sound here.
You can't do position control without position feedback.
You need position feedback. Your comments regarding why you're not going to use position feedback ("cumbersome and inaccurate due to vibration") are totally inaccurate. There is no vibration unless you don't mount the equipment correctly. It's not any more cumbersome than anything else.
If you want to do position control, you're going to put a position sensor somewhere.
How well you want to do position control, how accurate you want it to be, how long you want the startup/homing routine to be, those are all questions for you the designer to answer. Being lazy now and choosing to slap a limit switch on because you can't be bothered to actually look at rotary encoders means you get to wonder for the life of your robot arm whether the arm is actually in the position you think it is.
"Isn't there a way one could move a DC motor or stepper motor to a particular angle with IC/modules like l293d or other motor drivers."
You can command an incremental position change. You cannot verify the final position.