There are some well established methods for this. For example, estimating forest C stocks and fluxes is done for national C accounts which are used by the Intergovernmental Panel on Climate Change (IPCC) for global reporting.
The usual method is to measure the diameter of all trees in a defined area (a plot, e.g. like you suggest of 100 square meters), then use allometric equations to estimate either the volume or biomass of each tree, then convert this to C (which is simply 50% of the total dry biomass). This gives you the C stock of all trees in your plot, which can then be scaled up to the entire forest if you wish (you would need lots of plots to make a credible estimate).
Allometric equations vary between tree species due to their differences in shape and wood density. However the general proportion of C varies little between species, it is always close to 50% of biomass. It is better to use allometric equations specific to the tree species you are measuring but if none are available you can use a generic equations:
For live tree biomass, diameters of a sample of trees are measured and converted to biomass and carbon estimates using allometric biomass regression equations. Such equations exist for many forest types; some are species-specific, whereas others, particularly in the tropics, are more generic in nature (e.g., Alves et al., 1997; Brown, 1997; Schroeder et al., 1997). Cutting and weighing a sufficient number of trees to represent the size and species distribution in a forest to generate local allometric regression equations with high precision, particularly in complex tropical forests, is extremely time-consuming and costly and may be beyond the means of most projects. The advantage of using generic equations, stratified by ecological zones (e.g., dry, moist, and wet; see Brown, 1997), is that they tend to be based on a larger number of trees (Brown, 1997) and span a larger range of diameters; these factors increase the precision of the equations. A disadvantage is that the generic equations may not accurately reflect the true biomass of trees in the project.
If you already have wood volume data, which is often available for production forests, you can use a Biomass Expansion Factor to convert volume to biomass, and then to C.
Another method is to use lidar (laser scanning) to generate a 3D model of the trees in your plot and then calculate their volume -> biomass -> C.
Since trees are the easiest component of the forest to measure, there are methods to estimate the other C pools (litter, soil, roots, dead wood) because it is generally too difficult and time consuming to measure these. E.g. dead wood is generally 5-40% the C of live trees, but ideally you would use an estimate based on a similar forest type.
You specifically mention climax forest - though the method is the same for all forests it is likely to be less accurate for climax forests because they tend to be more variable in tree size and most allometrics have been developed for smaller trees (e.g. in plantations) and are usually not as accurate for larger trees. To improve accuracy you would need to cut down some trees and measure all of their components to derive your own allometrics.
BTW eucalypt and conifer forests in temperate regions have the highest C stocks per unit of land area of any forests.