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We study the fair allocation of indivisible goods among agents with identical, additive valuations but individual budget constraints. Here, the indivisible goods--each with a specific size and value--need to be allocated such that the bundle assigned to each agent is of total size at most the agent's budget. Since envy-free allocations do not necessarily exist in the indivisible goods context, compelling relaxations--in particular, the notion of envy-freeness up to $k$ goods (EFk)--have received significant attention in recent years. In an EFk allocation, each agent prefers its own bundle over that of any other agent, up to the removal of $k$ goods, and the agents have similarly bounded envy against the charity (which corresponds to the set of all unallocated goods). Recently, Wu et al. (2021) showed that an allocation that satisfies the budget constraints and maximizes the Nash social welfare is $1/4$-approximately EF1. However, the computation (or even existence) of exact EFk allocations remained an intriguing open problem. We make notable progress towards this by proposing a simple, greedy, polynomial-time algorithm that computes EF2 allocations under budget constraints. Our algorithmic result implies the universal existence of EF2 allocations in this fair division context. The analysis of the algorithm exploits intricate structural properties of envy-freeness. Interestingly, the same algorithm also provides EF1 guarantees for important special cases. Specifically, we settle the existence of EF1 allocations for instances in which: (i) the value of each good is proportional to its size, (ii) all goods have the same size, or (iii) all the goods have the same value. Our EF2 result extends to the setting wherein the goods' sizes are agent specific.