The only time that total costs could possibly equal fixed costs is if your firm has closed down. Total costs, by definition, are the sum or total fixed costs and total variable costs. So if total fixed costs are equal to total costs, then by definition total variable costs must be zero.
I can not think of any normal situation in which you could be producing anything and still have total variable costs of 0.
But if your firm were closed down, total costs would equal total fixed costs. This would be true because you would still have fixed costs, but you would not be making anything and so you would have no variable costs.
I guess you could argue that someone who works for themselves and produces a service could have 0 variable costs since they don't pay themselves a salary.
So imagine a barber who has a shop lit by the sun and who uses only scissors (maybe in some Third World country). That person would have only his fixed costs (chair, maybe, and scissors) no matter how many hair cuts he gave. But that's a pretty weird scenario.
In business an economics the composition and behavior of cost of production is expressed using different models. In one such model, which closely reflects the reality of cost behavior in many firms, the total cost of production is divided in two cost components - fixed cost and variable cost. The total cost is equal to fixed cost plus variable cost.
The fixed cost component is called fixed because it remains fixed irrespective of total quantity produced. This cost is incurred for items like rent for the land and building, depreciation, interest, and salary of top management. These cost are related to the production capacity created by the firm rather than the actual production. This fixed cost incurred by a firm remains same irrespective of the extent to which the production capacity is utilized. So it will be same irrespective of whether the production is zero or upto 100% production capacity.
The variable cost is directly proportional to the actual volume of production. If variables cost for one unit of production is x, then for N units of production it will be N times x. The variable costs are incurred for things like direct raw material and direct labour used for production,
From above discussion it becomes clear that a firm's total cost will be equal to fixed cost only when its production is zero, but the firm continues to exist and maintain its productive capacity.
If the firm closes down completely, rather than just stopping production for some period, then the firm will no longer have any capacity to produce even one unit of product. In such a case the fixed cost will also become zero. The company may have to bear some losses in closing down, but once it is closed down there are no costs of any type incurred.