Yes it is possible to compact without a plate compactor. But you must first know what factors affects compaction. The factors that affects compaction are soil type, moisture content, compactive energy and depth of the lift.
For soil with high cohesion like clay, silts and clayey silts are harder to compact as they have higher resistance of their particles. Yet, granular soil, like sand, gravel and other coarse soils are easier to compact as they have higher density.
Moisture content plays a great role to execute compacting any kind of soil. Soil moisture should be the right amount before compaction. To ensure that you have achieved the required water content, you must use the hand test. Take a part of the soil and try molding it, if it breaks and crumbles, it is still dry and you need to add more water. If the soil holds together, then you can now compact your entire area. And if you made it sticky, you need to dry it for a couple of time because it is too wet.
There are other tools you can use to compact such as a lawn roller, hand tamper of sledge hammer. You must achieve a consistent amount of pressure and speed of rolling if you are using a roller. You must tamper/roll efficiently so that you will be able to distribute the required compaction on your entire area. If you are working with a great depth, you can do layering so you can make sure that every area are well compacted.