They're components of a tensor density, as the names already indicate, e.g. "mass density" for $ρ$. The same is true of the Lagrangian density $$ that comprises the integrand of an action integral $\int d^4x$, where
$$d^4x = dx^0 ∧ dx^1 ∧ dx^2 ∧ dx^3,$$
assuming, here, we're speaking of 4D manifolds, like space-time.
The cardinal example is the canonical stress tensor associated with $$, which I will denote here $$, which may be written as:
$$^ρ_ν = \sum_{0≤A<n} \frac{∂}{∂v^A_ρ} v^A_ν - δ^ρ_ν , \quad v^A_ρ = ∂_ρ q^A,$$
for a field given in component form by $q(x) = \left(q^A(x): A = 0≤A<n\right)$, where $∂_ρ = ∂/∂x^ρ$. In here, and the following, I will use the summation convention (sum over repeated index pairs in monomials), so that the above sum may be written as:
$$\sum_{0≤A<n} \frac{∂}{∂v^A_ρ} v^A_ν ⇔ \frac{∂}{∂v^A_ρ} v^A_ν.$$
The Lagrangian may be written as a 4-form, while the components of the stress tensor density may be written as a 3-form as follows:
$$L = d^4x, \quad P_ν = ^ρ_ν ∂_ρ ˩ d^4x,$$
where the (bi-linear) contraction operator $˩$ is defined recursively on differential forms by
$$∂_ρ ˩ (a ∧ α) = a_ρ α - a ∧ (∂_ρ ˩ α), \quad ∂_ρ ˩ b = 0,$$
where $a = a_νdx^ν$ is a 1-form and $b$ a 0-form (i.e. scalar). Thus, for instance,
$$
∂_0 ˩ d^4x = +dx^1∧dx^2∧dx^3, \quad ∂_1 ˩ d^4x = -dx^0∧dx^2∧dx^3,\\
∂_2 ˩ d^4x = +dx^0∧dx^1∧dx^3, \quad ∂_3 ˩ d^4x = -dx^0∧dx^1∧dx^2.
$$
Thus, we could also write
$$\begin{align}
P_ν
&= \left(\frac{∂}{∂v^A_ρ} v^A_ν - δ^ρ_ν \right) ∂_ρ ˩ d^4x\\
&= v^A_ν \frac{∂}{∂v^A_ρ} ∂_ρ ˩ d^4x - δ^ρ_ν ∂_ρ ˩ d^4x\\
&= ∂_νq^A p_A - ∂_ν ˩ L,
\end{align}$$
where we define the 3-form $p_A$ conjugate to $q^A$ by:
$$p_A = \frac{∂}{∂v^A_ρ} ∂_ρ ˩ d^4x.$$
In turn, this is associated with the flow of a vector field $X = X^ν ∂_ν$ by
$$P_X ≡ X^ν P_ν = (Xq^A) p_A - X ˩ L.$$
You can think of this as the response of a deformation given locally in coordinate form by $X = Δx$, with $Δq^A$ given by the chain rule as $Δxq^A = Δx^ν ∂_νq^A$, and
$$P_{Δx} = Δq^A p_A - Δx ˩ L.$$
Generically, for a stress tensor density with components $^ρ_ν$, one can define:
$$T_ν = ^ρ_ν ∂_ρ ˩ d^4x, \quad T_X = X^ν T_ν = X^ν ^ρ_ν ∂_ρ ˩ d^4x.$$
The stress tensor which arises in contexts, such as general relativity or fluid dynamics, would then be expressed in terms of the canonical stress tensor as $^ρ_ν = ^ρ_ν + ∂_μ ^{ρμ}_ν$, where the components $^{ρμ}_ν = -^{μρ}_ν$ would be associated with a 2-form as:
$$p_ν = \frac{1}{2}^{ρμ}_ν ∂_μ ˩ ∂_ρ ˩ d^4x,$$
hence $T_ν = P_ν + dp_ν$, where the adjustment $dp_ν$ is given by the Belinfante correction or a generalization of it.
Expressing the stress tensor this way as a 3-form allows one to express the divergence of the stress tensor density as an exterior differential:
$$\begin{align}
dT_ν
&= d\left(^ρ_ν ∂_ρ ˩ d^4x\right)\\
&= ∂_μ^ρ_ν dx^μ ∧ \left(∂_ρ ˩ d^4x\right)\\
&= ∂_μ^ρ_ν δ^μ_ρ d^4x\\
&= ∂_ρ^ρ_ν d^4x.
\end{align}$$
Adopting coordinates $x^0 = t$ and $\left(x^1, x^2, x^3\right) = (x, y, z)$, and denoting the dimension of length, time duration and mass, respectively by $(L,T,M)$, of action by $H = ML^2/T$ and of $d^4x$ by $Ω = TL^3$, the Lagrangian 4-form, coordinate deformation, stress tensor 3-form, and their components will have the dimensions:
$$
[L] = H, \quad [] = \frac{H}{Ω} = \frac{M}{LT^2},\\
\left[Δx^ν\right] = [ν] ≡ \left[x^ν\right], \quad \left[∂_ν\right] = \frac{1}{[ν]}, \quad [Δx] = 1,\\
\left[T_{Δx}\right] = \frac{H}{Ω}, \quad \left[T_ν\right] = \frac{H}{Ω}\frac{1}{[ν]}, \quad \left[^ρ_ν\right] = \frac{H}{Ω}\frac{[ρ]}{[ν]},
$$
therefore, using $[0] = T$ and $[1] = [2] = [3] = L$, we have the following:
$$\begin{align}
\left[^0_0\right] &= \frac{H}{Ω}\frac{T}{T} = \frac{1}{L^3}\frac{ML^2}{T^2},\\
\left[^i_0\right] &= \frac{H}{Ω}\frac{L}{T} = \frac{1}{L^3}\frac{ML^2}{T^2}\frac{L}{T}\quad(i=1,2,3),\\
\left[^0_j\right] &= \frac{H}{Ω}\frac{T}{L} = \frac{1}{L^3}\frac{ML}{T}\quad(j=1,2,3),\\
\left[^i_j\right] &= \frac{H}{Ω}\frac{L}{L} = \frac{1}{L^3}\frac{ML}{T}\frac{L}{T}\quad(i,j=1,2,3),
\end{align}$$
which are, respectively, the dimensions for energy density, energy flux density, momentum density and momentum flux density (and pressure and stress).
