No one can be specific until they understand the context. You have three issues to clarify for a quality, helpful answer.
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Where can surface water runoff go? It's flat now so it goes nowhere, but if you sloped it gently like 1-2% grade, where would that slope put the water? Urban or rural? Are you on a hill or in a depression? Do you have existing drainage infrastructure nearby? E.g. ditches and storm sewers. Identify your neighbours surface water properties. Does their runoff flow onto your property or vice versa? Where is the water going and where should the water go.
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Can water penetrate the surface? You say it's overly compacted soil. What kind? Sand, clay, silty, loam? What is the surface? Grass? Shrubs? Options here include lawn areation. Preferably deep aeration. There are machines you can rent that are rollers covered in 1-2" hollow steel tubes that cut little plugs out of your lawn to reduce surface compaction and lets air and water in a little better. If your land is pooling, this won't be nearly enough, but its an essential step for part 3 below. Also plants, grass, shrubs help slow the flow of water, they absorb and transpire it and their root systems and annual cycles of decay turn mineral based ground into genuine soil. Soil's moisture absorbing qualities are defined primarily by its organic content within the matrix. Rich soil high in organic carbon looks like chocolate cake. It absorbs and drains well. If you have only mineral clay soils, you have the liner of a swimming pool. Either way, how deep is this surface layer? E.g. if its a water bowl for the first 2', but underneath the subsoil drains well, you can just ammend the surface and solve your problems.
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Where can water that penetrates the surface go? Take a 4' or longer length 1/4" steel tube and hammer it into the ground. Pull it up and check out the dirt plug you just made. What is the deeper soil made of. Sand drains well. Clay does not. Rock less so. Everything else is in between as per #2. You get little organic content the deeper you go because fewer roots can get there. Usually just big trees. Can water get into the spaces between particles? If not you need to fix this. But before you do, ask yourself, once the water get in this deep, where does it go? Creeks, rivers, drainage ditches, storm sewers. Your neighbour a urban triplex or a farmers field? What is their subsoil likely to be? How far down until bedrock?
Putting 1,2&3 together, a great, resilient system looks like this when rain falls.
A) gentle minimal rains land on vegetation like trees plants grass. The plants stop the water from even hitting the soil and it just evaporates off the improved surface area.
B) light rain does A, then trickles down to the surface where it is immediately absorbed by porous chocolate cake surface soils. Thirsty plants are happy. The soil both absorbs water and retains it so plants are happy between rains.
C) moderate rains first do A, then B, then C where the weight of more water on top pushes the water in the chocolate cake surface soils downward into the subsoil. Its well draining so its basically a sponge and can take huge amounts of water. It's also connected to neighbouring subsoils that also let water infiltrate and flow to a nearby creek, that then flows to anriver then on to a lake or ocean and its nobody' problem. Alternatively it goes through municipal storm sewers to the same end.
D) In very heavy or frequent rains, A,B,and C are maxed out and you have yet more water still do deal with. When this happens we are back to A. Surface runoff accumulates and now we are technically entering flood territory. A little surface run off is nothing to worry about. A lot is. Here is where your grading and elevation make all the difference. If the land is graded 1-2%, surface water flows downhill to whatever awaits it. Hopefully those storm sewers and creeks and watnot. If your property is on high ground you're in great shape relatively. You'll be the last to flood. If you are in the bottom of a hardpan bowl, put your swimsuit on, turn off the power and start emergency protocols.
If you managed to get through all this, feel free to give me your details and I can tailor some advice more specific to your situation. Feel free to dm me for specifics.
Edit: french drainage is one option, but you have to have the above context to know where the drains can put the water, or if there are better or cheaper options.
Source: lots of geotech in uni.