We don't know how far the 'Socrates' of the Republic, or of other Platonic dialogues represents the historical Socrates. Only saying.
As to the substance of your question, no-one leads a lonely and impoverished life in the Republic. There are three social classes (Rep. IV. passim): (1) the commercial and labouring class, who enjoy all the normal amenities of life. They can acquire money, marry, party, play sport and all other such things. They are excluded from political participation since they lack the requisite knowledge of the Forms, which are (roughly) the ultimate realities and values knowledge of which is essential to the correct conduct of politics.
Next we need to take account of (2) the auxiliaries (Rep. III.414B), who provide the military defence of the city (polis) and carry out the orders of (3) the Guardians who govern on the basis of insight vouchsafed by their knowledge of the Forms (Rep. VI passim).
The auxiliaries plainly cannot follow lives of self-indulgence; they need to observe the conditions necessary to fulfil their military role. But marriage is open to them as are the usual pleasures and enjoyments.
The Guardians are not 'lonely', since they form a cohesive, interacting group in the city as its governing elite. Nor are they 'impoverished' since the city makes full provision for their maintenance; all that is denied to them is, broadly speaking, personal property and marriage. They are not denied sex, indeed they are expected and required to have it in order to maintain the Guardian class over time. (Not all the Guardians' children will make the grade; 'inferior' offspring will be relegated to one of the two other classes just as some children of the other classes will be Guardian material and educated accordingly.) Guardian sex is, if I may put it so, collective; there are no permanent familial unions, there is only the necessary and presumably enjoyable activity of sexual intercourse after which each Guardian is assigned in due time her or his next sexual partner. The Guardians' children are brought up communally, none knowing who his or her parents are - any more than the parents know who their children are. All this is in the interests of Guardians' not having emotional attachments which deflect them from their pure devotion to public duty.
Who has sex with whom is strictly regulated during the Guardians' reproductive years (Rep. V.459D-E) but the controls are removed when the Guardians, female or male, have passed the age for producing children ((Rep. V.461B-C).
Life is demanding for the Guardians insofar as many of the normal forms of happiness are closed to them. Socrates realises that point; his main reply is that the Guardians have received a long and elaborate education at the end of which they have apprehended the ultimate nature of goodness and reality. This is an immense privilege and is itself so satisfying and transformative of their lives, raising them to such a plateau of enlightenment, that acceptance of the conditions imposed on them in governing the city is a moral obligation they owe in return to the city which has made provision for this education and its resulting enlightenment (Rep. VII.519C-520E).
Endnote on the average, normal person
Justice (dikaiosune), Plato tells us, is the health of the soul (Rep. IV.444C--445B). In regard to the social classes of the city (Rep.IV. 432B-434E) it obtains when each class performs its function, does its proper job (ergon). In the case of the Guardians, this means ruling. For the auxiliaries it consists in protecting the city and carrying out the orders of the Guardians. For average, normal people it means engaging in the ordinary business of life, in commerce or labour, not intruding in politics, and thus performing their proper function. As just, they enjoy the health of the soul : and to have a healthy soul is to be eudaimon - in a state of flourishing, well-being or happiness. So at least Plato holds.