This article deals with the fair allocation of indivisible goods and its generalization to matroids. The notions of fairness under consideration are equitability, proportionality and envy-freeness. It is long known that some instances fail to admit a fair allocation. However, an almost fair solution may exist if an appropriate relaxation of the fairness condition is adopted. This article deals with a matroid problem which comprises the allocation of indivisible goods as a special case. It is to find a base of a matroid and to allocate it to a pool of agents. We first adapt the aforementioned fairness concepts to matroids. Next we propose a relaxed notion of fairness said to be near to fairness. Near fairness respects the fairness up to one element. We show that a nearly fair solution always exists and it can be constructed in polynomial time in the general context of matroids.