For most beginner gardeners in an urban setting, the most affordable practical planter box is a fabric raised-bed planter, often sold as a fabric grow bed. A 2x4 ft fabric raised bed usually costs about $15–$30, holds enough soil for herbs, lettuce, peppers, or compact tomatoes, and is easy to move, store, and set up on a balcony, patio, or small yard. It is not a rigid wooden box, but it performs the same job for less money and gives better drainage and air circulation than many cheap plastic boxes.
If you specifically want a rigid planter box, the cheapest option is usually a basic plastic or resin rectangular planter, around $10–$25 for a 24–30 inch window-box style. It is fine for herbs and flowers, but it holds much less soil, so it is not the best value if you want to grow vegetables. A wooden DIY planter made from reclaimed boards can also be cheap, but you need tools, liner, and drainage holes, and untreated wood may not last as long.
For a beginner, I would choose a 2x4 ft fabric raised bed or two 10–15 gallon fabric grow bags for leafy greens and herbs. For tomatoes or peppers, use at least 20–30 gallons per plant. Buy the cheapest plain fabric bed you can find, then spend a little more on good potting mix and drainage. Place it where it gets 6–8 hours of sun, fill it with lightweight potting mix rather than garden soil, and check moisture daily because containers dry out faster than ground beds.
Bottom line: the most affordable urban planter box for a beginner is a fabric raised-bed planter, with a basic plastic rectangular planter as the cheapest rigid-box alternative. It keeps startup costs low while still giving you enough space to learn and grow.