Yes, a middle-class student can afford to study in BITS Pilani or Goa.

Though the course fees are a bit high for BITS Pilani or Goa there are scholarship opportunities for students based on their merit. Every student may enroll in a degree from an Indian prime institution thanks to several loan and scholarship options.

•  For students who obtain the chance to study at prestigious institutions like BITS, banks provide educational loans to students even at low-interest rates. You may use a bank's credit to study, and then once you graduate, you can pay back the bank.
•  Additionally, BITS provides its students with merit-based scholarships. According to a student's cumulative grade point average (CGPA) at the end of a certain semester, BITS offers many degrees of percentage tuition waivers, including 25%, 40%, and 75%.
•  BITS Pilani provides scholarships for its graduates and from other countries. For any of them to be granted to you during your term, you must apply through the necessary application processes.

• Scholarships for Merit in BITSAT (BITS Online Admission Test):

The first year's (2 semesters) tuition is waived for applicants who completed the BITSAT exams with outstanding scores. For the duration of the whole four-year program, a reduction in the hostel expenses is also provided. 

Yes, a student from a lower-middle-class background can afford to study at BITS Pilani or BITS Goa. There are several provisions of loans and scholarships for every student to afford a course from such a premier institute in India.

• Banks give educational loans to students even at low-interest rates for students who get chance of studying at premier institutes like BITS. You can study based on the credit given by banks and then repay the bank after their graduation.
• BITS also offers scholarships based on merit to its students. There are several levels of percentage waiver of tuition fees in BITS, including 25%, 40%, and 75%, depending on the CGPA of a student after a specific semester.
• BITS Pilani also offers the provision of international scholarships and scholarships from its own alumni. You need to apply to the required application processes for getting any of them during your tenure.

The video further explains how even a middle-class student can take admission in BITS

Therefore, it is absolutely possible for you to afford a study at institutes like BITS.