There are modifications of the RSA algorithm, which are considered suitable for long messages, after analyzing known games and attacks.
How to Strengthen the Security of RSA-OAEP, Boldyreva et al.
Your question addresses the block cipher concept. In the kernel, you have an algorithm, which operates on byte blocks of defined size; according to a defined operation mode, data blocks are preprocessed or post-processed. Easiest mode is the atomic ECB, "Electronic Codebook" mode, the one without ´knowledge´, i.e. you split your input stream 'plain.txt' in packets of defined size, encrypt them sequentially and concatenate the output in an output stream, e.g. 'encrypted.bin'. The transition from text to octets (=bytes) is in your case already anticipated, as you read a binary file. In other operation modes as CBC, CTR input or output from former blocks is an additional input for the block encryption.
These modes cover the structural aspect of fragmentation and memory on base of a byte input stream, which you start with as 'plain.txt'.
Now to the processing of blocks:
Usually a padding algorithm is applied to an input block, which adds salt and or mixes input bytes by hash functions, "OAEP" is a common candidate. Text book RSA omits padding.
Next step is transition from the padding output block to the numeric algorithm, the one you "put aside". This is defined PKCS#1 as mentioned, also the mapping back from number space to byte blocks.
You start with a RSA key length and associated modulus, which limits size of equivalent data blocks. Padding usually decreases the maximum size of user data, again. The evaluation of this usable block size is an essential input for the fragmentation.
Finally, decryption needs to know the length of the original stream. This must be encoded, too. E.g. you prefix the input stream with it's length.
Note: It's politically incorrect to apply block cipher modes to RSA, and some people will rather suggest you to limit your text file size. Listen to them, if you think, they solve your problem. :) Anyway, a stream will attract the block cipher concept.