This research aims to monitor the current situation of confinement in Spanish society motivated by COVID-19 crisis. For this, a study of its socio-family, psychological and educational impact is conducted. The sample (N = 165 families, 89.1% nuclear families with children living in the same household and 20.5% with a relative in a risk group) comes from the Aragonese region (Spain). The instruments used are: Beck-II Depression Inventory (BDI-II); Baron-Cohen and Wheelwright’s Empathy Quotient (EQ) with its cognitive empathy subscale, as well as an ad-hoc questionnaire, reviewed by a panel of experts, to learn about socio-personal, family and housing conditions, use of technology, involvement in school tasks and household, and working condition. The multiple regression analysis results show that the anxiety derived from the current situation is explained in 23.1% (p < 0.001) by the variables: gender (t = -2.31, p = 0.022), level of Internet consumption (t = 2.139, p = 0.034), increase of family conflicts (t = 2.980, p = 0.003) and help with school tasks (t = 2.980, p = 0.040). On the other hand, cognitive empathy is explained in 24.6% (p < 0.001) by the variables: gender (t = -4.690, p < 0.001) and mother’s hours of teleworking (t = 2.101, p = 0.037). All this leads us to conclusions related to preventive systems of social, psychological, and educational aspects to better serve families. These conclusions can be also be transferred to the future with an inclusive care to family settings from those three parameters.