Is it permissible for a wife to stay in her husband's family home, where five of his unmarried brothers also reside, while he is traveling, knowing that her own family's home is nearby but she is only allowed to stay there once a week?

If your staying overnight at your in-laws' house does not expose you to unlawful privacy (khalwa) or prohibited mixing (ikhtilat), then you must obey your husband in this matter and in refraining from going out unnecessarily except with his permission. However, if it exposes you to unlawful privacy or if there is a risk of fitna (temptation/strife), then it is not permissible for you to stay overnight with them, and you should come to an understanding with your husband to achieve what is beneficial and in accordance with the Shari'a.