Yes, if your have an extra SATA connector, then you can put a 3.5" HDD in the 5.25" DVD bay, but you will need either mounting brackets (figure 1 and 2) or a 3.5"-to-5.25" adapter (figure 3). If you use brackets, then you will also need a full 5.25" bezel or drive-bay cover (figure 4), and if you use an adapter, then you will need a 3.5" bezel (figure 5).
If you had built the system yourself, then the case would likely have come with the bezels/bay-covers since they don’t know what you would end up putting in the case, but a system from a manufacturer will usually not include them.
You can purchase them online (e.g., on eBay) for fairly cheap (<$5). You can also check with some small “mom and pop” computer shops in town to see if they have some old computers in the basement with them; they will often give them to you for free.
In a pinch, you can also install it without brackets or an adapter by screwing it directly to of the sides of the 5.25" bay (figure 6). However, because it doesn’t fit correctly, you will need to ensure that you screw it in tightly so that it isn’t loose and doesn’t jiggle. You will also want to put something under it to make it stable and hold it still (something firm but soft like stiff rubber is good). The top drive in the photo below is bad because it hangs in the air, the bottom one better because you can put something under it.
You will also want to use rubber washers (figure 7) to screw it in to reduce vibration and avoid it shaking the whole case, but this is true for all hard-drives, optical-drives, and anything that has moving parts that shake or spin.
Figure 1: Mounting brackets
Figure 2: 3.5" drive in mounting brackets
Figure 3: 3.5" to 5.25" adapter
Figure 4: 5.25" bezel/bay cover
Figure 5: 3.5" bezels
Figure 6: Photo of 3.5" drives installed in 5.25" bays without brackets or adapters
Figure 7: Photo of some rubber washers